Question

2. Trace minerals in drinking water affect the taste. Also, very high concentrations can be a health hazard. A study published in Environmental Studies (1982: pp 62-66), reports the zinc concentrations (in mg/L) at six locations. At each location, the concentration at surface water and bottom water were collected. The pairs of concentration at each location are given below Location Bottom Surface Difference 0.430 0.415 0.266 0.238 0.567 0.390 0.531 0.410 0.707 0.605 0.716 0.609 (a) Carry out a paired t test to determine whether there is strong evidence to suggest that the mean concentration of zinc at the bottom is greater than that at the surface. Use 0.05 significance level. (b) State your conclusions?

answer this using SAS program by ANOVA

Answer 1

Solution

Part (a)

Paired t-test

Let X = Zinc concentration (in mg/L) at the bottom of the location

      Y = Zinc concentration (in mg/L) at the surface of the location

And D = X – Y.

Then, D ~ N(µ, σ²) where σ² is unknown.

Claim:

Mean zinc concentration (in mg/L) at the bottom is greater than that at the surface.

Hypotheses:

Null: H₀: µ = µ₀= 0 Vs Alternative: H_A: µ > 0 [claim]

Test Statistic:

t = (√n) (Dbar - µ₀)/s where

Dbar and s are respectively, sample average and sample standard deviation based on n observations on X and Y.

Calculations

Summary of Excel calculations is given below:

Location (i)	xi	yi	di
1	0.43	0.415	0.015
2	0.266	0.238	0.028
3	0.567	0.39	0.177
4	0.531	0.41	0.121
5	0.707	0.605	0.102
6	0.716	0.609	0.107

n =	6
dbar	0.091667
s(d)	0.060688
µ0	0
tcal	3.699834
α =	0.05
tcrit =	2.015048
p-value =	0.007001

Distribution, Critical Value and p-value:

Under H₀, t ~ t_{n - 1}. Hence, for level of significance α%, Critical Value = upper α% point of

t_{n - 1} and p-value = P(t_{n - 1} > tcal).

Significance level: 5% [i.e., 0.05 - given]

Using Excel Functions: Statistical TINV TDIST, the above are found to be as given in the above table.

Decision:

Since tcal > tcrit, or equivalently, p-value < α, H₀ is rejected. Answer

Part (b)

Conclusion:

There is sufficient evidence to suggest that the claim is valid. i.e., Mean zinc concentration (in mg/L) at the bottom is greater than that at the surface. Answer

DONE